| L(s) = 1 | + 1.93·2-s + 0.539·3-s + 1.72·4-s + 1.04·6-s + 0.979·7-s − 0.523·8-s − 2.70·9-s − 0.751·11-s + 0.932·12-s + 0.473·13-s + 1.89·14-s − 4.46·16-s − 5.23·18-s − 1.84·19-s + 0.528·21-s − 1.45·22-s + 5.84·23-s − 0.282·24-s + 0.914·26-s − 3.07·27-s + 1.69·28-s + 1.36·29-s − 5.75·31-s − 7.58·32-s − 0.405·33-s − 4.68·36-s − 8.66·37-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 0.311·3-s + 0.864·4-s + 0.425·6-s + 0.370·7-s − 0.185·8-s − 0.903·9-s − 0.226·11-s + 0.269·12-s + 0.131·13-s + 0.505·14-s − 1.11·16-s − 1.23·18-s − 0.423·19-s + 0.115·21-s − 0.309·22-s + 1.21·23-s − 0.0576·24-s + 0.179·26-s − 0.592·27-s + 0.320·28-s + 0.252·29-s − 1.03·31-s − 1.34·32-s − 0.0705·33-s − 0.780·36-s − 1.42·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.93T + 2T^{2} \) |
| 3 | \( 1 - 0.539T + 3T^{2} \) |
| 7 | \( 1 - 0.979T + 7T^{2} \) |
| 11 | \( 1 + 0.751T + 11T^{2} \) |
| 13 | \( 1 - 0.473T + 13T^{2} \) |
| 19 | \( 1 + 1.84T + 19T^{2} \) |
| 23 | \( 1 - 5.84T + 23T^{2} \) |
| 29 | \( 1 - 1.36T + 29T^{2} \) |
| 31 | \( 1 + 5.75T + 31T^{2} \) |
| 37 | \( 1 + 8.66T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 43 | \( 1 - 8.60T + 43T^{2} \) |
| 47 | \( 1 - 0.285T + 47T^{2} \) |
| 53 | \( 1 - 7.91T + 53T^{2} \) |
| 59 | \( 1 - 8.41T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 7.78T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 5.17T + 79T^{2} \) |
| 83 | \( 1 - 3.43T + 83T^{2} \) |
| 89 | \( 1 + 8.38T + 89T^{2} \) |
| 97 | \( 1 + 2.58T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.33922615431827606529802845920, −6.78558931540032895355607006834, −5.80490243014843434528504215132, −5.48234939001033785887291281699, −4.70794970213191037449077042854, −4.01137024888450800859702189339, −3.16441595303156385082353869122, −2.69232653109671082608295284165, −1.65131740198782328185412878724, 0,
1.65131740198782328185412878724, 2.69232653109671082608295284165, 3.16441595303156385082353869122, 4.01137024888450800859702189339, 4.70794970213191037449077042854, 5.48234939001033785887291281699, 5.80490243014843434528504215132, 6.78558931540032895355607006834, 7.33922615431827606529802845920
